| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Multiply by polynomial |
| Difficulty | Standard +0.3 This is a straightforward application of the binomial expansion with n = -1/2, followed by a routine multiplication by a linear polynomial. Part (i) requires standard substitution and coefficient simplification, while part (ii) involves factoring out constants and multiplying expansions—both are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Either: Obtain correct (unsimplified) version of \(x\) or \(x^2\) term from \((1-4x)^{\frac{1}{2}}\) | M1 | |
| Obtain \(1 + 2x\) | A1 | |
| Obtain \(6x^2\) | A1 | |
| Or: Differentiate and evaluate \(f(0)\) and \(f'(0)\) where \(f'(x) = k(1-4x)^{-\frac{3}{2}}\) | M1 | |
| Obtain \(1 + 2x\) | A1 | |
| Obtain \(6x^2\) | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Combine both \(x^2\) terms from product of \(1 + 2x\) and answer from part (i) | M1 | |
| Obtain \(5\) | A1 | [2] |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **Either:** Obtain correct (unsimplified) version of $x$ or $x^2$ term from $(1-4x)^{\frac{1}{2}}$ | M1 | |
| Obtain $1 + 2x$ | A1 | |
| Obtain $6x^2$ | A1 | |
| **Or:** Differentiate and evaluate $f(0)$ and $f'(0)$ where $f'(x) = k(1-4x)^{-\frac{3}{2}}$ | M1 | |
| Obtain $1 + 2x$ | A1 | |
| Obtain $6x^2$ | A1 | [3] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Combine both $x^2$ terms from product of $1 + 2x$ and answer from part (i) | M1 | |
| Obtain $5$ | A1 | [2] |
---2 (i) Expand $\frac { 1 } { \sqrt { } ( 1 - 4 x ) }$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying the coefficients.\\
(ii) Hence find the coefficient of $x ^ { 2 }$ in the expansion of $\frac { 1 + 2 x } { \sqrt { } ( 4 - 16 x ) }$.
\hfill \mbox{\textit{CAIE P3 2012 Q2 [5]}}